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Revealed Preference
Hal R. Varian∗
January 2005
Revised: September 20, 2006
Abstract This is a survey of revealed preference analysis focusing on the period since Samuelson’s seminal development of the topic with emphasis on empirical applications. It was prepared for Samuelsonian Economics and the 21st Century, edited by Michael Szenberg.
1 Introduction
In January 2005 I conducted a search of JSTOR business and economics jour- nals for the phrase “revealed preference” and found 997 articles. A search of Google scholar returned 3,600 works that contained the same phrase. Surely, revealed preference must count as one of the most influential ideas in eco- nomics. At the time of its introduction it was a major contribution to the pure theory of consumer behavior, and the basic idea has been applied in a number of other areas of economics. In this essay I will briefly describe of the history of revealed preference, starting first descriptions of the concept in Samuelson’s papers. These papers subsequently stimulated a substantial amount of work devoted to refinements ∗Email contact: [email protected]
and extension of Samuelson’s ideas. This theoretical works, in turn, led to a literature on the use of revealed preference analysis for empirical work that is still growing rapidly.
2 The pure theory of revealed preference
Samuelson [1938] contains the first description of the concept he later called “revealed preference.” The initial terminology was “selected over.”^1 In this paper, Samuelson stated what has since become known as the “Weak Axiom of Revealed Preference” by saying “... if an individual selects batch one over batch two, he does not at the same time select two over one.” Let us state Samuelson’s definitions a bit more formally.
Definition 1 (Revealed Preference) Given some vectors of prices and chosen bundles (pt, xt) for t = 1,... , T , we say xt^ is directly revealed pre- ferred to a bundle x (written xtRDx) if ptxt^ ≥ ptx. We say xt^ is revealed preferred to x (written xtRx) if there is some sequence r, s, t,... , u, v such that prxr^ ≥ prxs, psxs^ ≥ psxt, · · · , puxu^ ≥ pux. In this case, we say the relation R is the transitive closure of the relation RD.
Definition 2 (Weak Axiom of Revealed Preference) If xtRDxs^ then it is not the case that xsRDxt. Algebraically, ptxt^ ≥ ptxs^ implies psxs^ < psxt.
Subsequently, building on the work of Little [1949], Samuelson [1948] sketched out an argument describing how one could use the revealed pref- erence relation to construct a set of indifference curves. This proof was for two goods only, and was primarily graphical. Samuelson recognized that a general proof for multiple goods was necessary, and left this as an open question. (^1) As Richter [1966] has pointed out, “selected over” has the advantage over “revealed preference” in that it avoids confusion about circular definition of “preference.” Unfortu- nately, the original terminology didn’t catch on.
and base revealed preference on pure set-theoretic arguments involving the completion of partial orders. This period culminated in the publication of Chipman et al. [1971], which contained a series of chapters that would seem to be the last word on revealed preference. Several years later Sondermann [1982], following Richter [1966]’s analysis, provided a one-paragraph proof of the basic revealed preference result, albeit a proof that used relatively sophisticated mathematics.
3 Afriat’s approach
Most of the theoretical work described above starts with a demand function: a complete description of what would be chosen at any possible budget. Afriat [1967] offered quite a different approach to revealed preference theory. He started with a finite set of observed prices and choices and asked how to actually construct a utility function that would be consistent with these choices.^2 The standard approach showed, in principle, how to construct preferences consistent with choices, but the actual preferences were described as limits or as a solution to some set of partial differential equations. Afriat’s approach, by contrast, was truly constructive, offering an explicit algorithm for to calculate a utility function consistent with the finite amount of data, whereas the other arguments were just existence proofs. This makes Afriat’s approach much more suitable as a basis for empirical analysis. Afriat’s approach was so novel that most researchers at the time did not recognize its value. In addition, Afriat’s exposition was not entirely transparent. Several years later Diewert [1973] offered a somewhat clearer exposition of Afriat’s main results. (^2) I once asked Samuelson whether he thought of revealed preference theory in terms of a finite or infinite set of choices. His answer, as I recall, was: “I thought of having a finite set of observations... but I always could get more if I needed them!”
4 From theory to data
During the late 1970s and early 1980s there was considerable interest in esti- mating aggregate consumer demand functions. Christensen, Lars, Jorgensen and Lau [1975] and Deaton [1983] are two notable examples. In reading this work, it occurred to me that it could be helpful to use revealed preference as a pre-test for this econometric analysis. After all, the Strong Axiom of Revealed Preference was a necessary and sufficient condition for data to be consistent with utility maximization. If the data satisfied SARP there would be some utility function consistent with the observations. If the data violated SARP, no such utility function would exist. So why not test those inequalities directly? I dug into the literature a bit and discovered that Koo [1963] had already thought of doing this, albeit with a somewhat different motivation. However, as Dobell [1965] pointed out, his analysis was not quite correct so there was still something left to be done. Furthermore I recognized the received theory, using WARP and SARP, was not well-suited for empirical work, since it was built around the assump- tion of single-valued demand functions. In 1977, during a visit to Berke- ley, Andreu Mas-Collel pointed me to Diewert [1973]’s exposition of Afriat’s analysis, which seemed to me to be a more promising basis for empirical applications. Diewert [1973] in turn led to Afriat [1967]. I corresponded with Afriat during this period, and he was kind enough to send me a package of his writing on the subject. His monograph Afriat [1987] offered the clearest exposition of his work in this area, though, as I discovered, it was not quite explicit enough to be programmed into a computer. I worked on reformulating Afriat’s argument in a way that would be directly amenable to computer analysis. While doing this, I recognized that Afriat’s condition of “cyclical consistency” was basically equivalent to Strong Axiom. Of course, in retrospect this had to be true since both cyclical
5 Consistency
Consistency is, of course, the central focus of the early work on revealed preference. As we have seen, several authors contributed to its solution, including Samuelson, Houthakker, Afriat and others. The most convenient result for empirical work, as I suggested above, comes from Afriat’s approach.
Definition 4 (Generalized Axiom of Revealed Preference) The data (pt, xt) satisfy the Generalized Axiom of Revealed Preference (GARP) if xtRxs implies psxs^ ≤ psxt.
GARP, as mentioned above, is equivalent to what Afriat called “cyclical consistency.” That the only difference between GARP and SARP is that the strong inequality in SARP becomes a weak inequality in GARP. This allows for multivalued demand functions and “flat” indifference curves, which turns out to be important in empirical work. Now we can state the main result.
Theorem 1 (Afriat’s Theorem.) Given some choice data (pt, xt) for t = 1 ,... , T , the following conditions are equivalent.
- There exists a nonsatiated utility function u(x) that rationalizes the data in the sense that for all t, u(xt) ≥ u(x) for all x such that ptxt^ ≥ ptx.
- The data satisfy GARP.
- There is a positive solution (ut, λt) to the set of linear inequalities
ut^ ≤ us^ + λsps(xt^ − xs)textf oralls, t.
- There exists a nonsatiated, continuous, monotone, and concave utility function u(x) that rationalizes the data.
This theorem offers two equivalent, testable conditions for the data to be consistent with utility maximization. The first is GARP, which, as we have seen, is a small generalization of Houthakker’s SARP. The second condition is whether there is a positive solution to a certain set of linear inequalities. This can easily be checked by linear programming methods. However, from the viewpoint of computational efficiency it is much easier just to check GARP. The only issue is to figure out how to compute the revealed preference relation in an efficient way. Let us define a matrix m that summarizes the direct revealed preference relation. In this matrix the (s, t) entry is given by mst = 1 if ptxt^ ≥ ptxs and mst = 0 otherwise. In order to test GARP all that is necessary is to compute the transitive closure of the relation summarized by this matrix. What algorithms are appropriate? Dobell [1965] recognized that this could be accomplished simply by taking the T th power of the T ×T binary matrix that summarizes the direct revealed preference relation. However, it turned out the computer scientists had a much more efficient algorithm. Warshall [1962] had shown a few years earlier how to use dynamic programming to compute the transitive closure in just T 3 steps. Combining the work of Afriat and Warshall effectively solved the problem of finding a computationally efficient method of testing data for consistency with utility maximization. One could simply construct the matrix summariz- ing the direct relation, compute the transitive closure and then check GARP.
5.1 Empirical analysis
Several authors have tested revealed preference conditions on different sorts of data. The “best” data, in some sense, is experimental data involving individual subjects since one can vary prices in such a setting and so test choice behavior over a wide range of environments.
interpretation of these findings is that Becker’s random choice model isn’t a very appealing alternative hypothesis. But, for all the criticism directed at the classical theory of consumer behavior there seem to be few alternative hypotheses other than Becker’s that can be applied using the same sorts of data used for revealed preference analysis.
5.2 Goodness of fit
It is of interest to consider ways to relax the revealed preference tests so that one might say “these data are almost consistent with GARP.” Afriat [1967] defines a “partial efficiency” measure which can be used to measure how well a given set of data satisfies utility maximization.
Definition 5 (Efficiency levels) We say that xt^ is directly revealed pre- ferred to x at efficiency level e if eptxt^ ≥ ptx.
We define the transitive closure of this relation as Re in the usual way. If e = 1 this is the standard direct revealed preference relation. If e = 0 nothing is directly revealed preferred to anything else, so GARP is vacuously satisfied. Hence there is some critical level e∗^ where the data just satisfy GARP. It is easy to find the critical level e by doing a binary search. Varian [1990] suggests defining et^ separately for each observation and then finding those et^ that are as close as possible to 1 (in some norm). I interpret these et^ as a “minimal perturbation.” They can be interpreted as error terms and thus be used to give a statistical interpretation to the goodness-of-fit measure. Whitney and Swofford [1987] suggest using the number of violations as a fit measure, while Famulari [1995] uses a measure which is roughly the fraction of violations that occur divided by the fraction that could have occurred. Houtman and Maks [1985] proposes computing the maximal subset of the data that is consistent with revealed preference. These measures are reviewed and compared in Gross [1995] who also offers his own suggestions.
6 Form
The issue of testing for various sorts of separability had been considered by Afriat in unpublished work and independently examined by Diewert and Parkan [1985]. The Diewert-Parkhan work extended the linear inequalities described in Afriat’s Theorem. They showed that if an appropriate set of lin- ear inequalities had positive solutions, then the data satisfied the appropriate form restriction. To get the flavor of this analysis, suppose that some observed data (pt, xt) were generated by a differentiable concave utility function u(x). Differentia- bility and concavity imply that
u(xt) ≤ u(xs) + Du(xt)(xs^ − xt) for all s, t
The first-order conditions for utility maximization imply
Du(xt) = λtpt^ for all t.
Putting these together, we find that a necessary condition for the data to be consistent with utility maximization is that there is a set of positive numbers (ut, λt), which can be interpreted as utility levels and marginal utilities of income, that satisfy the linear inequalities
ut^ ≤ us^ + λs(psxt^ − ptxt) for all s, t.
Furthermore, the existence of a solution to this set of inequalities is a sufficient condition as well. This can be proved by defining a utility function as the lower envelope of a set of hyperplanes defined as follows:
u(x) = min s us^ + λsps(x − xs).
Afriat [1967] had used a similar construction but went further showed that
GOOD 1
GOOD 2
Engel curve (linear) x x^1 x 3
2
Figure 1: GARP with homothetic preference.
erence (HARP) if for every sequence r, s, t,... , u, v
prxs prxr
psxt psxs^ · · ·^
puxv puxu^ ≥^1.
It turns out that there is an easy computation to check whether or not this condition is satisfied that uses methods that are basically the same as those in Warshall’s algorithm. Using similar methods, Browning [1984] came up with a nice test for life-cycle consumption models which rests on the constancy of the marginal utility of income in this framework. Subsequently Blundell et al. [2003] recognized that the logic used in the homotheticity tests could be extended to a much more general setting. Suppose one had estimates of Engel curves from other data. Then these Engel curves could be used to construct a set of data that could be subjected to revealed preference tests. The logic is the same as that described in Figure 1, but uses a an estimated Engel curve rather than the linear Engel curve implied by homotheticity. See Figure 2 for a simple example. The Blundell-Browning-Crawford approach is very useful for empirical work since cross-sectional household data can be used to estimate Engel curves, either parametrically or nonparametrically. See Blundell [2005] for further developments in this area.
GOOD 1
GOOD 2
Engel curve
Figure 2: GARP with arbitrary Engel curve.
Other restrictions on functional form, such as various forms of separabil- ity, have been examined by Varian [1982a]. Tests for expected utility max- imization and related models are described in Green and Srivastava [1986], Osbandi and Green [1991], Varian [1983], Varian [1988], Bar-Shira [1992].
7 Forecasting
Suppose we are given a finite set of observed budgets and choices (pt, xt) for t = 1,... , T that are consistent with GARP and a new price p^0 and expenditure y^0. What are the possible bundles x^0 that could be demanded at (p^0 , y^0 )? Clearly all that is necessary is to describe the set of x^0 for which the (expanded) data set (pt, xt) for t = 0,... , T satisfy GARP. Varian [1982a] calls this the set of supporting bundles. Figure 3 shows the geometry. In an analogous way, one can choose a new bundle x^0 and ask for the set of prices at which this bundle could be demanded. This is the set of supporting prices. Formally,
S(x^0 ) = {p^0 : (pt, xt) satisfy GARP for t = 0,... T }
Of course, one could also ask about demanded bundles or prices that are
x 2
x 1
P^1
x^1
x^0
RP(x^0 )
RW(x^0 )
Figure 4: RP (x^0 ) and RW (x^0 ): simple case.
preferred to x^0 (RP (x^0 )) and set of x’s that are revealed worse than x^0 (RW (x^0 ). A very simple example is shown in Figure 4. The possible set of supporting prices for x^0 must lie in the shaded cone so every such set of prices imply that x^0 is revealed preferred to the points in RW (x^0 ). Similarly, the points in the convex hull of the bundles revealed preferred to x^0 must themselves be preferred to x^0 for any concave utility function that rationalizes the data. Of course Figure 4 uses only one observation. As we get more observations on demand, we will get tighter bounds on RP (x^0 ) and RW (x^0 ), as shown in Figure 5. Another approach, also suggested by Varian [1982a] is to try to compute bounds on specific utility functions. A very convenient choice in this case is what Samuelson [1974] calls the money metric utility function. First define the expenditure function
e(p, u) = min pz such that u(z) ≥ u.
x 2
x 1
RP(x^0 )
RW(x^0 )
xo x 1
x 2 x 3
x 4 x 5
Figure 5: RP (x^0 ) and RW (x^0 ): a more complex case.
It is not hard to see that under minimal regularity conditions e(p, u) will be a strictly increasing function of u. Now define
m(p, x) = e(p, u(x)).
For fixed p, m(p, x) is a strictly increasing function of utility, so it is itself a utility function that represents the same preferences. Varian [1982a] suggested that given a finite set of data (pt, xt) one could define an upper bound to the money metric utility by using
m+(p, x) = min pzt^ such that ztRx.
Subsequently, Knoblauch [1992] showed that this bound was in fact tight: there were preferences that rationalized the observed choices that had m+(p, x) as their money-metric utility function. Varian [1982a] defined a lower bound to Samuelson’s money metric utility function and showed that it was tight. Of course, using restrictions on utility form such as HARP allow for tighter bounds. There are several papers on the implications of such re-
References
S. N. Afriat. The equivalence in two dimensions of the strong and weak axioms of revealed preference. Metroeconomica, 17:24–28, 1965.
Sydney Afriat. The construction of a utility function from expenditure data. International Economic Review, 8:67–77, 1967.
Sydney Afriat. On the constructibility of consistent price indices between several periods simultaneously. In Angus Deaton, editor, Essays in Theory and Measurement of Demand: in Honour of Sir Richard Stone. Cambridge University Press, Cambridge, England, 1981.
Sydney Afriat. Logic of Choice and Economic Theory. Clarendon Press, Oxford, 1987.
James Andreoni and John Miller. Giving according to GARP: An experi- mental test of the consistency of preferences for altruism. Econometrica, 70(2):737–753, 2002.
Ziv Bar-Shira. Nonparametric test of the expected utility hypothesis. Amer- ican Journal of Agricultural Economics, 74(3):523–533, 1992.
Raymond C. Battalio, John H. Kagel, Robin C. Winkler, Edwin B. Fisher, Robert L BAsmann, and Leonard Krasner. A test of csnsumer demand theory using observations of individual consumer purchases. Western Eco- nomic Journal, 11(4):411–428, 1873.
Gary Becker. Irrational behavior and economic theory. Journal of Political Economy, 70:1–13, 1962.
Richard Blundell. How revealing is revealed preference? European Economic Journal, 3:211–235, 2005.
Richard Blundell, Martin Browning, and I. Crawford. Nonparametric Engel curves and revealed preference. Econometrica, 71(1):205–240, 2003.
Stephen Bronars. The power of nonparametric tests. Econometrica, 55(3): 693–698, 1985.
Martin Browning. A non-parametric test of the life-cycle rational expecta- tions hypothesis. International Economic Review, 30:979–992, 1984.
J. S. Chipman, L. Hurwicz, M. K. Richter, and H. F. Sonnenschein. Prefer- ences, Utility and Demand. Harcourt Brace Janovich, New York, 1971.
Dale Christensen, Lars, Jorgensen and Lawrence Lau. Transcendental loga- rithmic utility functions. American Economic Review, 65:367–383, 1975.
James C. Cox. On testing the utility hypothesis. Technical report, University of Arizona, 1989.
Angus Deaton. Demand analysis. In Z. Griliches and M. Intrilligator, editors, Handbook of Econometrics. JAI Press, Greenwich, CT, 1983.
Erwin Diewert and Celick Parkan. Tests for consistency of consumer data and nonparametric index numbers. Journal of Econometrics, 30:127–147,
W. E. Diewert. Afriat and revealed preference theory. The Review of Eco- nomic Studies, 40(3):419–425, 1973.
A. R. Dobell. A comment on A. Y. C. Koo’s an empirical test of revealed preference theory. Econometrica, 33(2):451–455, 1965.
Steve Dowrick and John Quiggin. International comparisons of living stan- dards and tastes: A revealed-preference analysis. The American Economic Review, 84(1):332–341, 1994.
